\documentclass[a4paper,12pt]{article}
\title{how to program RSDC-GARCH}
\author{ZHU Cai\\ Department of Applied Mathematics}
\usepackage{amsmath,amsthm}

\begin{document}
\maketitle
\noindent\textbf{P} denotes the transition matrix of HMM:
\[
\begin{array}{ccc}

\end{array}
\]

\noindent $Y_t=\{y_t, y_{t-1}, y_{t-2}, \cdots\}$, history information set of data obtained through date t;\\

\noindent $f(y_t|s_t=j,Y_{t-1};\theta)$ denotes the conditional density of $y_t$;\\
\noindent $\eta_t$ is a $R\times1$ vector, and $\eta_t[j]= f(y_t|s_t=j,Y_{t-1};\theta)$\\

\noindent $\mathcal{P}(S_t=j|Y_t;\theta)$ denotes probability of HMM stays at the state j at time t, based on data
obtained through date t and based on knowledge of population paremeters $\theta$;\\
\noindent $\xi_{t|t}$ is a $R\times1$ vector, and $\xi_{t|t}[j]=\mathcal{P}(S_t=j|Y_t;\theta)$\\

\noindent $\mathcal{P}(S_{t+1}=j|Y_t;\theta)$ denotes probability of HMM stays at the state j at time t+1, based on data obtained through date t and based on knowledge of population paremeters $\theta$;\\
\noindent $\xi_{t+1|t}$ is a $R\times1$ vector, and $\xi_{t+1|t}[j]=\mathcal{P}(S_{t+1}=j|Y_t;\theta)$\\

\noindent $\mathcal{P}(S_t=j|Y_{t-1};\theta)$ denotes probability of HMM stays at the state j at time t, based on data obtained through date t and based on knowledge of population paremeters $\theta$;\\
\noindent $\xi_{t|t-1}$ is a $R\times1$ vector, and $\xi_{t|t-1}[j]=\mathcal{P}(S_t=j|Y_{t-1};\theta)$\\

\noindent At each date t, there are following equations:\\
\begin{equation}
\centering
\hat{\xi}_{t|t}=\frac{\hat{\xi}_{t|t-1}\odot\eta_t}{\textbf{1}^{'}(\hat{\xi}_{t|t-1}\odot\eta_t)}
\end{equation}
\begin{equation}
\centering
\hat{\xi}_{t+1|t}=\textbf{P}\cdot\hat{\xi}_{t|t}
\end{equation}



\end{document}
